3.180 \(\int \frac {\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=307 \[ \frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d}+\frac {\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac {b \left (15 a^4-20 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {\left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d} \]

[Out]

-2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^6/d+1/8*b*(15*a^4-20*a^2*b^2+8*b^4)*arct
anh(cos(d*x+c))/a^6/d-1/15*(23*a^4-35*a^2*b^2+15*b^4)*cot(d*x+c)/a^5/d-cot(d*x+c)*csc(d*x+c)/b/d+1/8*(8*a^4-9*
a^2*b^2+4*b^4)*cot(d*x+c)*csc(d*x+c)/a^4/b/d+1/2*a*cot(d*x+c)*csc(d*x+c)^2/b^2/d-1/30*(15*a^4-22*a^2*b^2+10*b^
4)*cot(d*x+c)*csc(d*x+c)^2/a^3/b^2/d+1/4*b*cot(d*x+c)*csc(d*x+c)^3/a^2/d-1/5*cot(d*x+c)*csc(d*x+c)^4/a/d

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Rubi [A]  time = 1.11, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2726, 3055, 3001, 3770, 2660, 618, 204} \[ -\frac {2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {\left (-35 a^2 b^2+23 a^4+15 b^4\right ) \cot (c+d x)}{15 a^5 d}+\frac {b \left (-20 a^2 b^2+15 a^4+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {\left (-22 a^2 b^2+15 a^4+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac {\left (-9 a^2 b^2+8 a^4+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6/(a + b*Sin[c + d*x]),x]

[Out]

(-2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*d) + (b*(15*a^4 - 20*a^2*b^2 + 8*
b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - ((23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^5*d) - (Cot[c + d*
x]*Csc[c + d*x])/(b*d) + ((8*a^4 - 9*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*b*d) + (a*Cot[c + d*x]
*Csc[c + d*x]^2)/(2*b^2*d) - ((15*a^4 - 22*a^2*b^2 + 10*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^3*b^2*d) + (b*
Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d)

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2726

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^6, x_Symbol] :> -Simp[(Cos[e + f*x]*(a
+ b*Sin[e + f*x])^(m + 1))/(5*a*f*Sin[e + f*x]^5), x] + (Dist[1/(20*a^2*b^2*m*(m - 1)), Int[((a + b*Sin[e + f
*x])^m*Simp[60*a^4 - 44*a^2*b^2*(m - 1)*m + b^4*m*(m - 1)*(m - 3)*(m - 4) + a*b*m*(20*a^2 - b^2*m*(m - 1))*Sin
[e + f*x] - (40*a^4 + b^4*m*(m - 1)*(m - 2)*(m - 4) - 20*a^2*b^2*(m - 1)*(2*m + 1))*Sin[e + f*x]^2, x])/Sin[e
+ f*x]^4, x], x] + Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*m*Sin[e + f*x]^2), x] + Simp[(a*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*m*(m - 1)*Sin[e + f*x]^3), x] - Simp[(b*(m - 4)*Cos[e + f*x]*(a +
b*Sin[e + f*x])^(m + 1))/(20*a^2*f*Sin[e + f*x]^4), x]) /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] &&
NeQ[m, 1] && IntegerQ[2*m]

Rule 3001

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx &=-\frac {\cot (c+d x) \csc (c+d x)}{b d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\int \frac {\csc ^4(c+d x) \left (4 \left (15 a^4-22 a^2 b^2+10 b^4\right )-2 a b \left (10 a^2-b^2\right ) \sin (c+d x)-10 \left (4 a^4-4 a^2 b^2+3 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{40 a^2 b^2}\\ &=-\frac {\cot (c+d x) \csc (c+d x)}{b d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\int \frac {\csc ^3(c+d x) \left (-30 b \left (8 a^4-9 a^2 b^2+4 b^4\right )-2 a b^2 \left (28 a^2+5 b^2\right ) \sin (c+d x)+8 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{120 a^3 b^2}\\ &=-\frac {\cot (c+d x) \csc (c+d x)}{b d}+\frac {\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\int \frac {\csc ^2(c+d x) \left (16 b^2 \left (23 a^4-35 a^2 b^2+15 b^4\right )-2 a b^3 \left (41 a^2-20 b^2\right ) \sin (c+d x)-30 b^2 \left (8 a^4-9 a^2 b^2+4 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{240 a^4 b^2}\\ &=-\frac {\left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}+\frac {\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\int \frac {\csc (c+d x) \left (-30 b^3 \left (15 a^4-20 a^2 b^2+8 b^4\right )-30 a b^2 \left (8 a^4-9 a^2 b^2+4 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{240 a^5 b^2}\\ &=-\frac {\left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}+\frac {\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^6}-\frac {\left (b \left (15 a^4-20 a^2 b^2+8 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^6}\\ &=\frac {b \left (15 a^4-20 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {\left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}+\frac {\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\left (2 \left (a^2-b^2\right )^3\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 d}\\ &=\frac {b \left (15 a^4-20 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {\left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}+\frac {\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\left (4 \left (a^2-b^2\right )^3\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 d}\\ &=-\frac {2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}+\frac {b \left (15 a^4-20 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {\left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}+\frac {\left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\\ \end {align*}

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Mathematica [A]  time = 1.41, size = 504, normalized size = 1.64 \[ \frac {736 a^5 \tan \left (\frac {1}{2} (c+d x)\right )-3 a^5 \sin (c+d x) \csc ^6\left (\frac {1}{2} (c+d x)\right )+41 a^5 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )-656 a^5 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+6 a^5 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )+15 a^4 b \csc ^4\left (\frac {1}{2} (c+d x)\right )-270 a^4 b \csc ^2\left (\frac {1}{2} (c+d x)\right )-15 a^4 b \sec ^4\left (\frac {1}{2} (c+d x)\right )+270 a^4 b \sec ^2\left (\frac {1}{2} (c+d x)\right )-1800 a^4 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1800 a^4 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-1120 a^3 b^2 \tan \left (\frac {1}{2} (c+d x)\right )-20 a^3 b^2 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+320 a^3 b^2 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+120 a^2 b^3 \csc ^2\left (\frac {1}{2} (c+d x)\right )-120 a^2 b^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )+2400 a^2 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2400 a^2 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-1920 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )-32 \left (23 a^5-35 a^3 b^2+15 a b^4\right ) \cot \left (\frac {1}{2} (c+d x)\right )+480 a b^4 \tan \left (\frac {1}{2} (c+d x)\right )-960 b^5 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+960 b^5 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{960 a^6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6/(a + b*Sin[c + d*x]),x]

[Out]

(-1920*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 32*(23*a^5 - 35*a^3*b^2 + 15*a*b^4
)*Cot[(c + d*x)/2] - 270*a^4*b*Csc[(c + d*x)/2]^2 + 120*a^2*b^3*Csc[(c + d*x)/2]^2 + 15*a^4*b*Csc[(c + d*x)/2]
^4 + 1800*a^4*b*Log[Cos[(c + d*x)/2]] - 2400*a^2*b^3*Log[Cos[(c + d*x)/2]] + 960*b^5*Log[Cos[(c + d*x)/2]] - 1
800*a^4*b*Log[Sin[(c + d*x)/2]] + 2400*a^2*b^3*Log[Sin[(c + d*x)/2]] - 960*b^5*Log[Sin[(c + d*x)/2]] + 270*a^4
*b*Sec[(c + d*x)/2]^2 - 120*a^2*b^3*Sec[(c + d*x)/2]^2 - 15*a^4*b*Sec[(c + d*x)/2]^4 - 656*a^5*Csc[c + d*x]^3*
Sin[(c + d*x)/2]^4 + 320*a^3*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 41*a^5*Csc[(c + d*x)/2]^4*Sin[c + d*x] -
20*a^3*b^2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 3*a^5*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 736*a^5*Tan[(c + d*x)/2]
- 1120*a^3*b^2*Tan[(c + d*x)/2] + 480*a*b^4*Tan[(c + d*x)/2] + 6*a^5*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(960
*a^6*d)

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fricas [A]  time = 1.04, size = 1079, normalized size = 3.51 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

[-1/240*(16*(23*a^5 - 35*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 - 80*(7*a^5 - 13*a^3*b^2 + 6*a*b^4)*cos(d*x + c)^3
- 120*((a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c
)^2)*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 + 2*(a*cos(d*x + c)*s
in(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2))*sin(d*x
+ c) - 15*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*
a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 15*(15*a^4*b - 20*a^2*b^3 + 8*b^5
+ (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(-1/2*
cos(d*x + c) + 1/2)*sin(d*x + c) + 240*(a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c) - 30*((9*a^4*b - 4*a^2*b^3)*cos(
d*x + c)^3 - (7*a^4*b - 4*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/((a^6*d*cos(d*x + c)^4 - 2*a^6*d*cos(d*x + c)^2
+ a^6*d)*sin(d*x + c)), -1/240*(16*(23*a^5 - 35*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 - 80*(7*a^5 - 13*a^3*b^2 +
6*a*b^4)*cos(d*x + c)^3 - 240*((a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^
2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c)))*sin(
d*x + c) - 15*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b -
20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 15*(15*a^4*b - 20*a^2*b^3 + 8*b
^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(-1
/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 240*(a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c) - 30*((9*a^4*b - 4*a^2*b^3)*c
os(d*x + c)^3 - (7*a^4*b - 4*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/((a^6*d*cos(d*x + c)^4 - 2*a^6*d*cos(d*x + c
)^2 + a^6*d)*sin(d*x + c))]

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giac [A]  time = 0.60, size = 490, normalized size = 1.60 \[ \frac {\frac {6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 660 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1080 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{5}} - \frac {120 \, {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} - \frac {1920 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{6}} + \frac {4110 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5480 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2192 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 660 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1080 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 480 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 70 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 40 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{5}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/960*((6*a^4*tan(1/2*d*x + 1/2*c)^5 - 15*a^3*b*tan(1/2*d*x + 1/2*c)^4 - 70*a^4*tan(1/2*d*x + 1/2*c)^3 + 40*a^
2*b^2*tan(1/2*d*x + 1/2*c)^3 + 240*a^3*b*tan(1/2*d*x + 1/2*c)^2 - 120*a*b^3*tan(1/2*d*x + 1/2*c)^2 + 660*a^4*t
an(1/2*d*x + 1/2*c) - 1080*a^2*b^2*tan(1/2*d*x + 1/2*c) + 480*b^4*tan(1/2*d*x + 1/2*c))/a^5 - 120*(15*a^4*b -
20*a^2*b^3 + 8*b^5)*log(abs(tan(1/2*d*x + 1/2*c)))/a^6 - 1920*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(pi*floor(1/
2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*a^6) + (
4110*a^4*b*tan(1/2*d*x + 1/2*c)^5 - 5480*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 + 2192*b^5*tan(1/2*d*x + 1/2*c)^5 - 66
0*a^5*tan(1/2*d*x + 1/2*c)^4 + 1080*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 - 480*a*b^4*tan(1/2*d*x + 1/2*c)^4 - 240*a^
4*b*tan(1/2*d*x + 1/2*c)^3 + 120*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 70*a^5*tan(1/2*d*x + 1/2*c)^2 - 40*a^3*b^2*t
an(1/2*d*x + 1/2*c)^2 + 15*a^4*b*tan(1/2*d*x + 1/2*c) - 6*a^5)/(a^6*tan(1/2*d*x + 1/2*c)^5))/d

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maple [B]  time = 0.21, size = 629, normalized size = 2.05 \[ -\frac {b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d \,a^{2}}+\frac {b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{3}}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}}{8 d \,a^{4}}+\frac {6 b^{2} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,a^{2} \sqrt {a^{2}-b^{2}}}-\frac {15 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{2}}-\frac {2 \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \sqrt {a^{2}-b^{2}}}+\frac {b^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{5}}-\frac {b^{2}}{24 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b^{4}}{2 d \,a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{64 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {b^{3}}{8 d \,a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b^{5} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{6}}+\frac {b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{2}}-\frac {9 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{3}}+\frac {9 b^{2}}{8 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b}{4 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5 b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{4}}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}-\frac {11}{16 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{160 d a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 a d}+\frac {7}{96 d a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 a d}+\frac {2 \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right ) b^{6}}{d \,a^{6} \sqrt {a^{2}-b^{2}}}-\frac {6 \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right ) b^{4}}{d \,a^{4} \sqrt {a^{2}-b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6/(a+b*sin(d*x+c)),x)

[Out]

6/d*b^2/a^2/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-15/8/d/a^2*b*ln(tan(1/2*d
*x+1/2*c))-2/d/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+1/2/d/a^5*b^4*tan(1/2*
d*x+1/2*c)-1/24/d/a^3/tan(1/2*d*x+1/2*c)^3*b^2-1/2/d/a^5/tan(1/2*d*x+1/2*c)*b^4+1/64/d/a^2*b/tan(1/2*d*x+1/2*c
)^4+1/8/d/a^4*b^3/tan(1/2*d*x+1/2*c)^2-1/d/a^6*b^5*ln(tan(1/2*d*x+1/2*c))-1/64/d/a^2*b*tan(1/2*d*x+1/2*c)^4+1/
24/d/a^3*b^2*tan(1/2*d*x+1/2*c)^3-1/8/d/a^4*tan(1/2*d*x+1/2*c)^2*b^3+1/4/d/a^2*b*tan(1/2*d*x+1/2*c)^2-9/8/d/a^
3*b^2*tan(1/2*d*x+1/2*c)+9/8/d/a^3/tan(1/2*d*x+1/2*c)*b^2-1/4/d/a^2*b/tan(1/2*d*x+1/2*c)^2+5/2/d/a^4*b^3*ln(ta
n(1/2*d*x+1/2*c))+11/16/a/d*tan(1/2*d*x+1/2*c)-11/16/a/d/tan(1/2*d*x+1/2*c)+1/160/d/a*tan(1/2*d*x+1/2*c)^5-1/1
60/d/a/tan(1/2*d*x+1/2*c)^5-7/96/d/a*tan(1/2*d*x+1/2*c)^3+7/96/d/a/tan(1/2*d*x+1/2*c)^3+2/d/a^6/(a^2-b^2)^(1/2
)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))*b^6-6/d/a^4/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2
*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))*b^4

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 7.13, size = 1099, normalized size = 3.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^6/(a + b*sin(c + d*x)),x)

[Out]

tan(c/2 + (d*x)/2)^5/(160*a*d) + (tan(c/2 + (d*x)/2)^2*(b/(32*a^2) + (b*(7/(32*a) - b^2/(8*a^3)))/a))/d - (tan
(c/2 + (d*x)/2)*(b^2/(8*a^3) - 11/(16*a) + (2*b*(b/(16*a^2) + (2*b*(7/(32*a) - b^2/(8*a^3)))/a))/a))/d - (tan(
c/2 + (d*x)/2)^3*(7/(96*a) - b^2/(24*a^3)))/d - (b*tan(c/2 + (d*x)/2)^4)/(64*a^2*d) - (log(tan(c/2 + (d*x)/2))
*((15*a^4*b)/8 + b^5 - (5*a^2*b^3)/2))/(a^6*d) + (tan(c/2 + (d*x)/2)^2*((7*a^4)/3 - (4*a^2*b^2)/3) - a^4/5 - t
an(c/2 + (d*x)/2)^4*(22*a^4 + 16*b^4 - 36*a^2*b^2) + tan(c/2 + (d*x)/2)^3*(4*a*b^3 - 8*a^3*b) + (a^3*b*tan(c/2
+ (d*x)/2))/2)/(32*a^5*d*tan(c/2 + (d*x)/2)^5) - (atan((((-(a + b)^5*(a - b)^5)^(1/2)*((8*a^12 - 16*a^6*b^6 +
44*a^8*b^4 - 39*a^10*b^2)/(4*a^10) + ((2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^12 - 32*a^10*b^2))/(4*a^9))*(-(a +
b)^5*(a - b)^5)^(1/2))/a^6 + (tan(c/2 + (d*x)/2)*(31*a^10*b - 32*a^4*b^7 + 96*a^6*b^5 - 98*a^8*b^3))/(4*a^9))
*1i)/a^6 + ((-(a + b)^5*(a - b)^5)^(1/2)*((8*a^12 - 16*a^6*b^6 + 44*a^8*b^4 - 39*a^10*b^2)/(4*a^10) - ((2*a^2*
b - (tan(c/2 + (d*x)/2)*(24*a^12 - 32*a^10*b^2))/(4*a^9))*(-(a + b)^5*(a - b)^5)^(1/2))/a^6 + (tan(c/2 + (d*x)
/2)*(31*a^10*b - 32*a^4*b^7 + 96*a^6*b^5 - 98*a^8*b^3))/(4*a^9))*1i)/a^6)/((15*a^10*b - 8*b^11 + 44*a^2*b^9 -
99*a^4*b^7 + 113*a^6*b^5 - 65*a^8*b^3)/(2*a^10) + (tan(c/2 + (d*x)/2)*(16*a^10 - 8*b^10 + 42*a^2*b^8 - 94*a^4*
b^6 + 110*a^6*b^4 - 66*a^8*b^2))/(2*a^9) - ((-(a + b)^5*(a - b)^5)^(1/2)*((8*a^12 - 16*a^6*b^6 + 44*a^8*b^4 -
39*a^10*b^2)/(4*a^10) + ((2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^12 - 32*a^10*b^2))/(4*a^9))*(-(a + b)^5*(a - b)^
5)^(1/2))/a^6 + (tan(c/2 + (d*x)/2)*(31*a^10*b - 32*a^4*b^7 + 96*a^6*b^5 - 98*a^8*b^3))/(4*a^9)))/a^6 + ((-(a
+ b)^5*(a - b)^5)^(1/2)*((8*a^12 - 16*a^6*b^6 + 44*a^8*b^4 - 39*a^10*b^2)/(4*a^10) - ((2*a^2*b - (tan(c/2 + (d
*x)/2)*(24*a^12 - 32*a^10*b^2))/(4*a^9))*(-(a + b)^5*(a - b)^5)^(1/2))/a^6 + (tan(c/2 + (d*x)/2)*(31*a^10*b -
32*a^4*b^7 + 96*a^6*b^5 - 98*a^8*b^3))/(4*a^9)))/a^6))*(-(a + b)^5*(a - b)^5)^(1/2)*2i)/(a^6*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{6}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**6/(a+b*sin(d*x+c)),x)

[Out]

Integral(cot(c + d*x)**6/(a + b*sin(c + d*x)), x)

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